Optimal. Leaf size=109 \[ -\frac{256 \left (a+b x^2\right )^{11/4}}{77 a^4 c (c x)^{11/2}}+\frac{64 \left (a+b x^2\right )^{7/4}}{7 a^3 c (c x)^{11/2}}-\frac{8 \left (a+b x^2\right )^{3/4}}{a^2 c (c x)^{11/2}}+\frac{2}{a c (c x)^{11/2} \sqrt [4]{a+b x^2}} \]
[Out]
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Rubi [A] time = 0.126085, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{256 \left (a+b x^2\right )^{11/4}}{77 a^4 c (c x)^{11/2}}+\frac{64 \left (a+b x^2\right )^{7/4}}{7 a^3 c (c x)^{11/2}}-\frac{8 \left (a+b x^2\right )^{3/4}}{a^2 c (c x)^{11/2}}+\frac{2}{a c (c x)^{11/2} \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Int[1/((c*x)^(13/2)*(a + b*x^2)^(5/4)),x]
[Out]
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Rubi in Sympy [A] time = 15.7196, size = 95, normalized size = 0.87 \[ \frac{2}{a c \left (c x\right )^{\frac{11}{2}} \sqrt [4]{a + b x^{2}}} - \frac{8 \left (a + b x^{2}\right )^{\frac{3}{4}}}{a^{2} c \left (c x\right )^{\frac{11}{2}}} + \frac{64 \left (a + b x^{2}\right )^{\frac{7}{4}}}{7 a^{3} c \left (c x\right )^{\frac{11}{2}}} - \frac{256 \left (a + b x^{2}\right )^{\frac{11}{4}}}{77 a^{4} c \left (c x\right )^{\frac{11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(c*x)**(13/2)/(b*x**2+a)**(5/4),x)
[Out]
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Mathematica [A] time = 0.0624786, size = 63, normalized size = 0.58 \[ -\frac{2 \sqrt{c x} \left (7 a^3-12 a^2 b x^2+32 a b^2 x^4+128 b^3 x^6\right )}{77 a^4 c^7 x^6 \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((c*x)^(13/2)*(a + b*x^2)^(5/4)),x]
[Out]
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Maple [A] time = 0.008, size = 53, normalized size = 0.5 \[ -{\frac{2\,x \left ( 128\,{b}^{3}{x}^{6}+32\,a{b}^{2}{x}^{4}-12\,{a}^{2}b{x}^{2}+7\,{a}^{3} \right ) }{77\,{a}^{4}}{\frac{1}{\sqrt [4]{b{x}^{2}+a}}} \left ( cx \right ) ^{-{\frac{13}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(c*x)^(13/2)/(b*x^2+a)^(5/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{5}{4}} \left (c x\right )^{\frac{13}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^(5/4)*(c*x)^(13/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.20905, size = 77, normalized size = 0.71 \[ -\frac{2 \,{\left (128 \, b^{3} x^{6} + 32 \, a b^{2} x^{4} - 12 \, a^{2} b x^{2} + 7 \, a^{3}\right )}}{77 \,{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{c x} a^{4} c^{6} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^(5/4)*(c*x)^(13/2)),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(c*x)**(13/2)/(b*x**2+a)**(5/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{5}{4}} \left (c x\right )^{\frac{13}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^(5/4)*(c*x)^(13/2)),x, algorithm="giac")
[Out]